semiprimean

Semiprimean is a mathematical neologism used to denote the arithmetic mean of the two prime factors of a semiprime. A semiprime is a natural number that is the product of either two primes p and q (p ≤ q) or the square of a prime p^2. For a semiprime n = p q, the semiprimean is defined as s(n) = (p + q)/2. In the case n = p^2, s(n) = p.

Properties: The semiprimean s(n) lies between p and q, and equals p when p = q. It is an integer whenever p and q are both odd or equal; if one factor is 2 and the other odd, s(n) is a half-integer. Given n and s(n), the original primes can be recovered (p and q are the roots of t^2 − 2 s(n) t + n = 0), so p = s(n) − sqrt(s(n)^2 − n) and q = s(n) + sqrt(s(n)^2 − n), provided s(n)^2 − n is a perfect square.

Examples: For n = 15 = 3×5, s(15) = 4. For n = 21 = 3×7, s(21) = 5. For n = 4 = 2×2, s(4) = 2. For n = 14 = 2×7, s(14) = 4.5.

Etymology and usage: Semiprimean is a coined term formed from semiprime and the suffix -an, used in some expository contexts to highlight the mean of the prime factors. It is not part of standard number theory terminology, and many authors refer to the same concept more explicitly as the mean of the prime factors of a semiprime.

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